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Quadratic Forms

J.H. Conway, Contemporary Mathematics, Universal Quadratic Forms and the Fifteen Theorem, here.

The representation theory of quadratic forms has a long history, start- ing in the seventeenth century with Fermat’s assertions of 1640 about the numbers represented by x2 + y2. In the next century, Euler gave proofs of these and some similar assertions about other simple binary quadratics, and although these proofs had some gaps, they contributed greatly to setting the theory on a firm foundation. Lagrange started the theory of universal quadratic forms in 1770 by proving his celebrated Four Squares Theorem, which in current language is expressed by saying that the form x2+y2+z2+t2 is universal. The eighteenth century was closed by a considerably deeper statement – Legendre’s Three Squares Theorem of 1798; this found exactly which numbers needed all four squares. In his Theorie des Nombres of 1830, Legendre also created a very general theory of binary quadratics.

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